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A070932 Possible number of units in a (commutative or non-commutative) ring. 2
0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 36, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is a list of the numbers of units in R where R ranges over all commutative or non-commutative rings.

By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers - 1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.

Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^n-q^{n-1} for n >= 1 and q a prime power (see Rains link).

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. M. Rains, Comments on A070932

MATHEMATICA

max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &] - 1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* Jean-Fran├žois Alcover, Sep 10 2013 *)

PROG

(PARI) list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n->n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P-1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)), , 8); u=List(); for(i=3, #v, for(j=i, #v, P=v[i]*v[j]; if(P>lim, break); if(!vecsearch(v, P), listput(u, P))))); v \\ Charles R Greathouse IV, Jan 08 2013

CROSSREFS

Cf. A000010, A002202, A000252, A000961, A181062, A221178.

A000252 is a subsequence.

The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.

Sequence in context: A275804 A141825 A238369 * A161577 A093686 A155763

Adjacent sequences:  A070929 A070930 A070931 * A070933 A070934 A070935

KEYWORD

nonn,nice

AUTHOR

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 06 2013, Jan 08 2013

STATUS

approved

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Last modified June 28 17:16 EDT 2017. Contains 288839 sequences.